A significant consideration which must be faced by financial institutions (and individual investors) is the potential risk of future losses which is inherent in a given financial position, such as a portfolio. There are various ways for measuring potential future risk which are used under different circumstances. One commonly accepted measure of risk is the value at risk (“VAR”) of a particular financial portfolio. The VAR of a portfolio indicates the portfolio's market risk at a given percentile. In other words, the VAR is the greatest possible loss that the institution may expect in the portfolio in question with a certain given degree of probability during a certain future period of time. For example, a VAR equal to the loss at the 99th percentile of risk indicates that there is only a 1% chance that the loss will be greater than the VAR during the time frame of interest.
Generally, financial institutions maintain a certain percentage of the VAR in reserve as a contingency to cover possible losses in the portfolio in a predetermined upcoming time period. It is important that the VAR estimate be accurate. If an estimate of the VAR is too low, there is a possibility that insufficient finds will be available to cover losses in a worst-case scenario. Overestimating the VAR is also undesirable because funds set aside to cover the VAR are not available for other uses.
To determine the VAR for a portfolio, one or more models which incorporate various risk factors are used to simulate the price of each instrument in the portfolio a large number of times using an appropriate model. The model characterizes the price of the instrument on the basis of one or more risk factors, which can be broadly considered to be a market factor which is derived from tradable instruments and which can be used to predict or simulate the changes in price of a given instrument. The risk factors used in a given model are dependent on the type of financial instrument at issue and the complexity of the model. Typical risk factors include implied volatilities, prices of underlying stocks, discount rates, loan rates, and foreign exchange rates. Simulation involves varying the value of the risk factors in a model and then using the model to calculate instrument prices in accordance with the selected risk factor values. The resulting price distributions are aggregated to produce a value distribution for the portfolio. The VAR for the portfolio is determined by analyzing this distribution.
A particular class of instrument which is simulated is an option. Unlike simple securities, the price of an option is dependant upon the price of the underlying asset price, the volatility of changes in the underlying asset price, and possibly changes in various other option parameters, such as the time for expiration. (The price dependencies for other derivative instruments can be similarly complex.) An option can be characterized according to its strike price and the date it expires. The volatility of the option price is related to both of these factors. Sensitivity of the option volatility to these effects are commonly referred to skew and term. Measures of the volatility for a set of options can be combined to produce a volatility surface. For example, FIG. 1 is a graph of the implied volatility surface for S&P 500 index options on Sep. 27, 1995 as a function of strike level and term to expiration.
The volatility surface can be used to extract volatility values for a given option during simulation. The extracted volatility value is applied to an option pricing model which provides simulated option prices. These prices can be analyzed to make predictions about risk, such as the VAR of a portfolio containing options. The volatility surface is not static, but changes on a day-to-day basis. Thus, in order to make risk management decisions and for other purposes, changes in the volatility surface need to be simulated as well.
Various techniques can be used to simulate the volatility surface over time. In general financial simulations, two simulation techniques are conventionally used: parametric simulation and historical simulation. Variations of these techniques can be applied to simulate volatilities.
In a parametric simulation, the change in value of a given factor is modeled according to a stochastic or random function responsive to a noise component ε. During simulation, a suitable volatility surface can be used to extract a starting volatility value for the options to be simulated and this value then varied in accordance with randomly selected values of noise over the course of a simulation.
Although parametric simulation is flexible and permits the model parameters to be adjusted to be risk neutral, conventional techniques utilize a normal distribution for the random noise variations. As a result, probability distribution “fat-tails” which occur in real life must be explicitly modeled to compensate for the lack of this feature in the normal distribution. In addition, cross-correlations between various factors must be expressly represented in a variance-covariance matrix. The correlations between factors can vary depending on the circumstances and detecting these variations and compensating for them is difficult and can greatly complicate the modeling process. Moreover, the computational cost of determining the cross-correlations grows quadratically with the number of factors making it difficult to process models with large numbers of factors.
An alternative to parametric simulation is historical simulation. In a historical simulation, a historical record of data is analyzed to determine the actual factor values and these values are then selected at random during simulation. This approach is extremely simple and can accurately capture cross-correlations, volatilities, and fat-tail event distributions. However, this method is limited because the statistical distribution of values is restricted to the specific historical sequence which occurred. In addition, historical data may be missing or non-existent, particularly for newly developed instruments or risk factors, and the historical simulation is generally not risk neutral.
In addition to typical simulation based on an analysis of normal market performance, it is often useful to “stress test” a simulation model to determine the effect of very rare market occurrences on, e.g., the behavior of a portfolio. The events modeled during stress testing are atypical and it is generally not desired to represent them in the distribution of values used during a normal simulation process.
In a conventional method, the value of a market observable variable is changed by some large amount and than the impact of this change on the portfolio is simulated. The value of risk factors, such as implied volatility, can also be adjusted upwards or downwards by some amount. Multiple scenarios, typically chosen arbitrarily and from historical data gathered over 10 to 20 years, are often used to determine the reaction of a portfolio to such rare events. For example, a simulation could introduce a drop in the value of the S&P 500 and an increase in the implied volatility as observed for “Black Monday” of October, 1987 to determine the effect of a similar market on a portfolio at issue.
Although such conventional simulation techniques provide some insight into portfolio performance, the manner in which the volatility is adjusted is simplistic. In particular, conventional stress testing adjusts the value of all points on the respective implied volatility surface in parallel, raising or lowering the volatility surface by some fixed amount. In many situations, however, the actual behavior of the volatility surface is more complex. For example, the volatility of an option on a security may increase significantly for short term options but only a small degree for long term options.
Because conventional stress testing makes only gross changes to the surface values as a whole, scenarios like those in the example are modeled improperly and the simulation can over or underestimate significant amounts of risk.
Accordingly, there is a need to provide a better way to represent changes in implied volatility and other risk factors during stress test simulation of a portfolio.